The classical theory for modulational analysis which describes the long-time evolution and stability of fast oscillatory solutions
of nonlinear dispersive partial differential equations (PDEs) is based on the so-called modulational Ansatz for solutions
in the form of a weakly nonlinear narrowband wave train. In the context of gravity water waves,
the leading nontrivial terms give rise to the NLS equation for the evolution of the slowly varying wave envelope
(see [1]} for the original derivation in the irrotational case).
[2]} later proposed a higher-order approximation for waves on deep water,
which has since been extended to many other settings.
The Dysthe equation and its variants are widely used in the oceanographic community because of their efficiency and ability
to describe waves of moderately large steepness. It has been observed that numerical solutions of the Dysthe model
provide a better agreement with laboratory experiments than NLS and are able e.g.
to emulate the asymmetry of propagating wave packets, a feature not captured by NLS [3]}, [4]}.
